Composite Transformations

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Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Algebra II EOC · 2013 · Item 10 · 1 point

Worked example

Graph g has vertex (1, 2). Graph h has the same shape and vertex (−3, 5). Write h(x) in terms of g(x).

1. Horizontal change: −3 − 1 = −4, so the graph moves left 4.
2. A left shift of 4 is written inside as $g(x+4)$.
3. Vertical change: 5 − 2 = 3, so add 3 outside.
4. Check: $h(-3)=g(1)+3=2+3=5$.

$h(x)=g(x+4)+3$

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TEKSForge practice · unscored

1. 1. Describe the graph of $h(x)=g(x-2)-6$.

   Show answer and reasoning

   The graph of g moves right 2 and down 6.

   Subtracting 2 inside moves the graph right. Subtracting 6 outside lowers every output by 6.

2. 2. Graph h is graph g shifted left 5 and down 1. Write h(x).

   Show answer and reasoning

   $h(x)=g(x+5)-1$

   A left shift appears as adding inside the function; a downward shift appears as subtracting outside.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 10 · 1 point

The graph of function $h$ was obtained from the graph of function $g$ using a composite transformation, as shown. Which equation can be used to describe $h(x)$ in terms of $g(x)$?

1. $h(x) = g(x + 4) + 2$
2. $h(x) = g(x + 4) - 2$
3. $h(x) = g(x - 4) - 2$
4. $h(x) = g(x - 4) + 2$

Show STAAR answer and explanation

$h(x) = g(x - 4) - 2$

Each point on g moves 4 units right and 2 units down to reach h. A right shift of 4 uses $g(x-4)$, and a downward shift of 2 subtracts 2 outside the function.

$h(x) = g(x + 4) + 2$: $x+4$ shifts left, and adding 2 shifts up; both directions are reversed.

$h(x) = g(x + 4) - 2$: The vertical shift is correct, but $g(x+4)$ moves the graph left rather than right.

$h(x) = g(x - 4) + 2$: The horizontal shift is correct, but adding 2 moves the graph up rather than down.

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Archived released STAAR Algebra II EOC · 2013 · Item 29 · 1 point

For the functions $h$ and $g$, which statement is true if $h(x) = g(x + 14) - 12$?

1. The graph of $h$ is the result of the graph of $g$ being translated right 14 units and down 12 units.
2. The graph of $h$ is the result of the graph of $g$ being translated left 14 units and down 12 units.
3. The graph of $h$ is the result of the graph of $g$ being translated right 14 units and up 12 units.
4. The graph of $h$ is the result of the graph of $g$ being translated left 14 units and up 12 units.

Show STAAR answer and explanation

The graph of $h$ is the result of the graph of $g$ being translated left 14 units and down 12 units.

Adding 14 inside the function, $g(x+14)$, shifts the graph left 14 units. Subtracting 12 outside the function shifts it down 12 units.

The graph of $h$ is the result of the graph of $g$ being translated right 14 units and down 12 units.: $x+14$ inside the function shifts left, not right.

The graph of $h$ is the result of the graph of $g$ being translated right 14 units and up 12 units.: Both directions are reversed: $+14$ inside moves left and $-12$ outside moves down.

The graph of $h$ is the result of the graph of $g$ being translated left 14 units and up 12 units.: Subtracting 12 outside the function moves the graph down, not up.

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